Solitary pulses and periodic wave trains in a bistable FitzHugh-Nagumo model with cross diffusion and cross advection.

We describe analytically, and simulate numerically, traveling waves with oscillatory tails in a bistable, piecewise-linear reaction-diffusion-advection system of the FitzHugh-Nagumo type with linear cross-diffusion and cross-advection terms of opposite signs. We explore the dynamics of two wave types, namely, solitary pulses and their infinite sequences, i.e.

Nonlinear waves in a quintic FitzHugh-Nagumo model with cross diffusion: Fronts, pulses, and wave trains.

We study a tristable piecewise-linear reaction-diffusion system, which approximates a quintic FitzHugh-Nagumo model, with linear cross-diffusion terms of opposite signs. Basic nonlinear waves with oscillatory tails, namely, fronts, pulses, and wave trains, are described. The analytical construction of these waves is based on the results for the bistable case [Zemskov et al.

Periodic propagating waves coordinate RhoGTPase network dynamics at the leading and trailing edges during cell migration.

Migrating cells need to coordinate distinct leading and trailing edge dynamics but the underlying mechanisms are unclear. Here, we combine experiments and mathematical modeling to elaborate the minimal autonomous biochemical machinery necessary and sufficient for this dynamic coordination and cell movement. RhoA activates Rac1 via DIA and inhibits Rac1 via ROCK, while Rac1 inhibits RhoA through PAK.

Oscillatory multipulsons: Dissipative soliton trains in bistable reaction-diffusion systems with cross diffusion of attractive-repulsive type.

One-dimensional localized sequences of bound (coupled) traveling pulses, wave trains with a finite number of pulses, are described in a piecewise-linear reaction-diffusion system of the FitzHugh-Nagumo type with linear cross-diffusion terms of opposite signs. The simplest case of two bound pulses, the paired-pulse waves (pulse pairs), is solved analytically. The solutions contain oscillatory tails in the wave profiles so that the pulse pairs consist of a double-peak core and wavy edges.

Multifront regime of a piecewise-linear FitzHugh-Nagumo model with cross diffusion.

Oscillatory reaction-diffusion fronts are described analytically in a piecewise-linear approximation of the FitzHugh-Nagumo equations with linear cross-diffusion terms, which correspond to a pursuit-evasion situation. Fundamental dynamical regimes of front propagation into a stable and into an unstable state are studied, and the shape of the waves for both regimes is explored in detail. We find that oscillations in the wave profile may either be negligible due to rapid attenuation or noticeable if the damping is slow or vanishes.

Oscillatory pulse-front waves in a reaction-diffusion system with cross diffusion.

We explore traveling waves with oscillatory tails in a bistable piecewise linear reaction-diffusion system of the FitzHugh-Nagumo type with linear cross diffusion. These waves differ fundamentally from the standard simple fronts of the kink type. In contrast to kinks, the waves studied here have a complex shape profile with a front-back-front (a pulse-front) pattern.

Oscillatory pulses and wave trains in a bistable reaction-diffusion system with cross diffusion.

We study waves with exponentially decaying oscillatory tails in a reaction-diffusion system with linear cross diffusion. To be specific, we consider a piecewise linear approximation of the FitzHugh-Nagumo model, also known as the Bonhoeffer-van der Pol model. We focus on two types of traveling waves, namely solitary pulses that correspond to a homoclinic solution, and sequences of pulses or wave trains, i.

Wavy fronts in a hyperbolic FitzHugh-Nagumo system and the effects of cross diffusion.

We study a hyperbolic version of the FitzHugh-Nagumo (also known as the Bonhoeffer-van der Pol) reaction-diffusion system. To be able to obtain analytical results, we employ a piecewise linear approximation of the nonlinear kinetic term. The hyperbolic version is compared with the standard parabolic FitzHugh-Nagumo system.

Control of the G-protein cascade dynamics by GDP dissociation inhibitors.

A network of the Rho family GTPases, which cycle between inactive GDP-bound and active GTP-bound states, controls key cellular processes, including proliferation and migration. Activating and deactivating GTPase transitions are controlled by guanine nucleotide exchange factors (GEFs), GTPase activating proteins (GAPs) and GDP dissociation inhibitors (GDIs) that sequester GTPases from the membrane to the cytoplasm. Here we show that a cascade of two Rho family GTPases, RhoA and Rac1, regulated by RhoGDI1, exhibits distinct modes of the dynamic behavior, including abrupt, bistable switches, excitable overshoot transitions and oscillations.

The topology design principles that determine the spatiotemporal dynamics of G-protein cascades.

Small monomeric G-proteins control cellular behavior, cycling between inactive GDP-bound and active GTP-bound states. Activating and deactivating transitions are regulated by guanine nucleotide exchange factors (GEFs) and GTPase activating proteins (GAPs), respectively. G-proteins can control different GEF and GAP activities, thereby creating GTPase signaling cascades.

Long-range signaling by phosphoprotein waves arising from bistability in protein kinase cascades.

A hallmark of protein kinase/phosphatase cascades, including mitogen-activated protein kinase (MAPK) pathways, is the spatial separation of their components within cells. The top-level kinase, MAP3K, is phosphorylated at the cell membrane, and cytoplasmic kinases at sequential downstream levels (MAP2K and MAPK) spread the signal to distant targets. Given measured protein diffusivity and phosphatase activities, signal propagation by diffusion would result in a steep decline of MAP2K activity and low bisphosphorylated MAPK (ppMAPK) levels near the nucleus, especially in large cells, such as oocytes.